Heat 2 sticks and 2 tablespoons of butter in a saucepan over medium heat, until browned but not burnt (about 3-4 minutes), stirring frequently to break up browned bits. Remove and set aside in a large mixing bowl to cool for a minute or so. Meanwhile, add the flour, salt, baking powder, and baking soda to a small bowl and whisk to combine.
Add the brown and white sugars to the brown butter, whisking vigorously to avoid clumps, until combined. Next, add the eggs and vanilla extract to the butter and sugar mixture. Mix to combine until smooth (it will be thick).
Slowly add your dry ingredients (flour, baking soda, baking powder, salt) to the wet ingredients using a silicone spatula. Make sure to scrape the sides and bottom of the bowl to avoid dry clumps.
Once the dough is formed, chill it in the fridge covered, for 30 minutes. While the dough is chilling, preheat the oven to 350° F.
Using scoop (or hands) split the cookie dough into 30 equally-sized balls. Place on a cookie sheet lined with parchment paper. In a small bowl, mix the reserved 3 tablespoons of white sugar and 1 tablespoon of brown sugar. One by one, roll each dough ball in the sugar before placing back on the cookie sheet.
Bake the brown butter sugar cookies for 8 minutes, until the cookies are a pale golden color. You'll want to bake them in batches (they won't all fit on a cookie sheet). I suggest freezing some for later!
When the cookies are finished baking, top them with flaky sea salt (optional, but recommended). Let the cookies cool for 5 minutes and enjoy!